A Gary-Stu is considered to be a "proxy" or wish-fulfillment for the author; the term basically describes when a male author writes a book in which the main character is a version of his own self, just perfect/better (this is not true self-insertion); however, they usually serve as partners for Mary-Sues in a book written by a female. A Gary-Stu is seen as too flawless/perfect to be interesting or relatable, and can be a poorly developed character. Gary-Stus are what the male author wishes they could be/what the female author wishes they could have as a partner: special, beautiful, loved by all, perfect, etc. etc.

There are times when an author may just do true self-insertion. When an author does such, the character isn't usually considered a Gary-Stu; rather, the character is simply the author himself, unchanged.

Some hints that a character may be a Gary-Stu:
~ Exotic hair/eye color
~ Magic/super powers (Even if it is in a fantasy world, the Gary-Stu may have some extra special, super-uber cool power that's extremely uncommon/rare, and can wipe out an entire population)
~ Best at whatever he does
~ Unusually tragic past (i.e. Both parents died, whole family died, one parent left, etc. etc.)

Note, however, that sometimes in stories, features such as these are actually common, so some characters aren't really Gary-Stus.

Some other clues that a character may be a Gary-Stu:
~ Gary-Stu gets his true love and/or any girl he wants, who may be extremely attractive/wealthy/cool/perfect (in some cases, just like him)
~ Gray-Stu shares similar features/behavior with the author
~ Gary-Stu will always win in fights against his enemies
~ Gary-Stu is one of the most knowledgeable and clever people, and cannot be bested
~ Gray-Stu may be a part of the "in-crowd," or is welcomed into a group where he is practically worshipped, occasionally after being an outcast
~ Gray-Stu is taken care of, and has to worry about little, since usually everything works out for him (this usually only applies to Mary-Sues, since generally the female authors who write themselves as Mary-Sues are weak and wish only to be adored, while male authors may wish their own Gary-Stu to be viewed dominant above all men)

Still, remember that not all of these features indicate that a character is a Gary-Stu; perhaps just a few.

In many books, the main character is special in his own way, which is why the book focuses on him. Just remember that a Gray-Stu seems to be too perfect in some aspects, and has too much luck in some scenarios, thus, causing him to seem one-dimensional and blah.

*This information came from both Wikipedia, and from information I've gleaned from books I've read/read reviews about, where the main character is widely considered to be a Gary-Stu/Mary-Sue.